Question

discrete math

Write a recursive definition that generates the terms of each of the two following integer sequences: 9. a. 1,-1,2,-2, 4,-4,8,-8, 16,-16, b 1,2, 3,6, 11, 20, 37, 68, 125, 230, 423,. Hint: A recursive definition, which comprises initial conditions and a recurrence relation, is discussed in Section 5.6 of our textbook ] 10. Use a truth table to determine whether the following argument form is valid. Include a sentence or two referring to your truth table to support your answer

Answer 1

9)

a.

1, -1, 2, -2, 4, -4, 8, -8, 16, -16 .........

Following is the recursive definition,

$a_{n}=\begin{cases} a_{n-2}-a_{n-1},& \text{ if } n\,is\,even \\ -a_{n-1},& \text{ if } n\,is\,odd \end{cases}$

with base cases as

$a_{0}=1,\,\,a_{1}=-1$

b.

Following is the recursive definition,

$a_{n}=a_{n-1}+a_{n-2}+a_{n-3}$

with base cases as

$a_{0}=1,\,\,a_{1}=2,\,\,a_{2}=3$

10)

$p\rightarrow q$

$q\rightarrow r$

$\therefore \,\sim r\rightarrow \sim p$

$(p\rightarrow q)\wedge (q\rightarrow r)\,\,implies\,\,(p\rightarrow r)$

This property is called Hypothetical Syllogism.

Also Contrapositive of a logical statement is logically equivalent to the original statement

Taking the contrapositive we get

$(\sim r\rightarrow \sim p)$

So

$(\sim r\rightarrow \sim p) \,\,and\,\,(p\rightarrow r)\,\,are \,\, logically\,\,equivalent$

TTF TFF T F T FTT TF F FTF FTF TFT T 7

As we can see from the truth table, the conclusion  $p\rightarrow r$ is true for all possible input combinations of p,q,r. So, argument is valid.

And because  $p\rightarrow r$ and $(\sim r\rightarrow \sim p)$ are logically equivalent , the argument is valid for $(\sim r\rightarrow \sim p)$ as well.